The Rendering Equation
Direct (local) illumination
Light directly from light sources
No shadows Indirect (global) illumination
Hard and soft shadows
Diffuse interreflections (radiosity)
Glossy interreflections (caustics)
CS348B Lecture 13 Pat Hanrahan, Spring 2010
Radiosity
CS348B Lecture 13 Pat Hanrahan, Spring 2010
Page 1 Lighting Effects
Hard Shadows Soft Shadows
Caustics Indirect Illumination
CS348B Lecture 13 Pat Hanrahan, Spring 2010
Challenge
To evaluate the reflection equation the incoming radiance must be known
To evaluate the incoming radiance the reflected radiance must be known
CS348B Lecture 13 Pat Hanrahan, Spring 2010
Page 2 To The Rendering Equation
Questions 1. How is light measured? 2. How is the spatial distribution of light energy described? 3. How is reflection from a surface characterized? 4. What are the conditions for equilibrium flow of light in an environment?
CS348B Lecture 13 Pat Hanrahan, Spring 2010
The Grand Scheme
Light and Radiometry
Energy Balance
Surface Volume Rendering Equation Rendering Equation
Radiosity Equation
CS348B Lecture 13 Pat Hanrahan, Spring 2010
Page 3 Energy Balance
Balance Equation
Accountability [outgoing] - [incoming] = [emitted] - [absorbed] Macro level The total light energy put into the system must equal the energy leaving the system (usually, via heat).
Micro level The energy flowing into a small region of phase space must equal the energy flowing out.
CS348B Lecture 13 Pat Hanrahan, Spring 2010
Page 4 Surface Balance Equation
[outgoing] = [emitted] + [reflected]
CS348B Lecture 13 Pat Hanrahan, Spring 2010
Direction Conventions
BRDF Surface vs. Field Radiance
CS348B Lecture 13 Pat Hanrahan, Spring 2010
Page 5 Surface Balance Equation
[outgoing] = [emitted] + [reflected] + [transmitted]
BTDF
CS348B Lecture 13 Pat Hanrahan, Spring 2010
Two-Point Geometry
Ray Tracing
CS348B Lecture 13 Pat Hanrahan, Spring 2010
Page 6 Coupling Equations
Invariance of radiance
CS348B Lecture 13 Pat Hanrahan, Spring 2010
The Rendering Equation
Directional form
Transport operator Integrate over i.e. ray tracing hemisphere of directions
CS348B Lecture 13 Pat Hanrahan, Spring 2010
Page 7 The Rendering Equation
Surface form
Geometry term Integrate over all surfaces
Visibility term
CS348B Lecture 13 Pat Hanrahan, Spring 2010
The Radiosity Equation
Assume diffuse reflection 1. 2.
CS348B Lecture 13 Pat Hanrahan, Spring 2010
Page 8 Integral Equations
Integral Equations
Integral equations of the 1st kind
Integral equations of the 2nd kind
CS348B Lecture 13 Pat Hanrahan, Spring 2010
Page 9 Linear Operators
Linear operators act on functions like matrices act on vectors
They are linear in that
Types of linear operators
CS348B Lecture 13 Pat Hanrahan, Spring 2010
Solving the Rendering Equation
Rendering Equation
Solution
CS348B Lecture 13 Pat Hanrahan, Spring 2010
Page 10 Formal Solution
Neumann series
Verify
CS348B Lecture 13 Pat Hanrahan, Spring 2010
Successive Approximations
Successive approximations
Converged
CS348B Lecture 13 Pat Hanrahan, Spring 2010
Page 11 Successive Approximation
CS348B Lecture 13 Pat Hanrahan, Spring 2010
Paths
Page 12 Light Path
CS348B Lecture 13 Pat Hanrahan, Spring 2010
Light Path
CS348B Lecture 13 Pat Hanrahan, Spring 2010
Page 13 Light Paths
CS348B Lecture 13 Pat Hanrahan, Spring 2010
Light Transport
Integrate over all paths of all lengths
Question:
How to sample space of paths?
CS348B Lecture 13 Pat Hanrahan, Spring 2010
Page 14 Classic Ray Tracing
From Heckbert
Forward (from eye): E S* (D|G) L CS348B Lecture 13 Pat Hanrahan, Spring 2010
Photon Paths
Radiosity
Caustics
From Heckbert
CS348B Lecture 13 Pat Hanrahan, Spring 2010
Page 15 How to Solve It?
Finite element methods
Classic radiosity
Mesh surfaces
Piecewise constant basis functions
Solve matrix equation
Not practical for rendering equation Monte Carlo methods
Path tracing (distributed ray tracing)
Bidirectional ray tracing
Photon mapping
CS348B Lecture 13 Pat Hanrahan, Spring 2010
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